LMFDB - Embedded newform 1089.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1,1089)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.69570878012$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +2.61803 q^{5} -3.00000 q^{7} +2.23607 q^{8} -1.61803 q^{10} -1.76393 q^{13} +1.85410 q^{14} +1.85410 q^{16} +1.61803 q^{17} -5.85410 q^{19} -4.23607 q^{20} -3.47214 q^{23} +1.85410 q^{25} +1.09017 q^{26} +4.85410 q^{28} +4.47214 q^{29} +2.85410 q^{31} -5.61803 q^{32} -1.00000 q^{34} -7.85410 q^{35} +0.236068 q^{37} +3.61803 q^{38} +5.85410 q^{40} -11.9443 q^{41} -6.23607 q^{43} +2.14590 q^{46} -1.61803 q^{47} +2.00000 q^{49} -1.14590 q^{50} +2.85410 q^{52} +9.61803 q^{53} -6.70820 q^{56} -2.76393 q^{58} -10.3262 q^{59} -7.85410 q^{61} -1.76393 q^{62} -0.236068 q^{64} -4.61803 q^{65} -9.56231 q^{67} -2.61803 q^{68} +4.85410 q^{70} +5.56231 q^{71} +3.23607 q^{73} -0.145898 q^{74} +9.47214 q^{76} -9.47214 q^{79} +4.85410 q^{80} +7.38197 q^{82} +0.708204 q^{83} +4.23607 q^{85} +3.85410 q^{86} -0.527864 q^{89} +5.29180 q^{91} +5.61803 q^{92} +1.00000 q^{94} -15.3262 q^{95} -14.0344 q^{97} -1.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 3 q^{5} - 6 q^{7} - q^{10} - 8 q^{13} - 3 q^{14} - 3 q^{16} + q^{17} - 5 q^{19} - 4 q^{20} + 2 q^{23} - 3 q^{25} - 9 q^{26} + 3 q^{28} - q^{31} - 9 q^{32} - 2 q^{34} - 9 q^{35}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 3 * q^5 - 6 * q^7 - q^10 - 8 * q^13 - 3 * q^14 - 3 * q^16 + q^17 - 5 * q^19 - 4 * q^20 + 2 * q^23 - 3 * q^25 - 9 * q^26 + 3 * q^28 - q^31 - 9 * q^32 - 2 * q^34 - 9 * q^35 - 4 * q^37 + 5 * q^38 + 5 * q^40 - 6 * q^41 - 8 * q^43 + 11 * q^46 - q^47 + 4 * q^49 - 9 * q^50 - q^52 + 17 * q^53 - 10 * q^58 - 5 * q^59 - 9 * q^61 - 8 * q^62 + 4 * q^64 - 7 * q^65 + q^67 - 3 * q^68 + 3 * q^70 - 9 * q^71 + 2 * q^73 - 7 * q^74 + 10 * q^76 - 10 * q^79 + 3 * q^80 + 17 * q^82 - 12 * q^83 + 4 * q^85 + q^86 - 10 * q^89 + 24 * q^91 + 9 * q^92 + 2 * q^94 - 15 * q^95 + q^97 + 2 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	2.61803	1.17082	0.585410	−	0.810737i	$-0.300933\pi$
$5$	2.61803	1.17082	0.585410	+	0.810737i	$0.300933\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−1.61803	−0.511667
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.76393	−0.489227	−0.244613	−	0.969621i	$-0.578661\pi$
$13$	−1.76393	−0.489227	−0.244613	+	0.969621i	$0.578661\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	0	0
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	−4.23607	−0.947214
$21$	0	0
$22$	0	0
$23$	−3.47214	−0.723990	−0.361995	−	0.932180i	$-0.617904\pi$
$23$	−3.47214	−0.723990	−0.361995	+	0.932180i	$0.617904\pi$
$24$	0	0
$25$	1.85410	0.370820
$26$	1.09017	0.213800
$27$	0	0
$28$	4.85410	0.917339
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.85410	0.512612	0.256306	−	0.966596i	$-0.417495\pi$
$31$	2.85410	0.512612	0.256306	+	0.966596i	$0.417495\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−7.85410	−1.32759
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	3.61803	0.586923
$39$	0	0
$40$	5.85410	0.925615
$41$	−11.9443	−1.86538	−0.932691	−	0.360677i	$-0.882546\pi$
$41$	−11.9443	−1.86538	−0.932691	+	0.360677i	$0.882546\pi$
$42$	0	0
$43$	−6.23607	−0.950991	−0.475496	−	0.879718i	$-0.657731\pi$
$43$	−6.23607	−0.950991	−0.475496	+	0.879718i	$0.657731\pi$
$44$	0	0
$45$	0	0
$46$	2.14590	0.316395
$47$	−1.61803	−0.236015	−0.118007	−	0.993013i	$-0.537651\pi$
$47$	−1.61803	−0.236015	−0.118007	+	0.993013i	$0.537651\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−1.14590	−0.162054
$51$	0	0
$52$	2.85410	0.395793
$53$	9.61803	1.32114	0.660569	−	0.750765i	$-0.270315\pi$
$53$	9.61803	1.32114	0.660569	+	0.750765i	$0.270315\pi$
$54$	0	0
$55$	0	0
$56$	−6.70820	−0.896421
$57$	0	0
$58$	−2.76393	−0.362922
$59$	−10.3262	−1.34436	−0.672181	−	0.740387i	$-0.734642\pi$
$59$	−10.3262	−1.34436	−0.672181	+	0.740387i	$0.734642\pi$
$60$	0	0
$61$	−7.85410	−1.00561	−0.502807	−	0.864398i	$-0.667699\pi$
$61$	−7.85410	−1.00561	−0.502807	+	0.864398i	$0.667699\pi$
$62$	−1.76393	−0.224020
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−4.61803	−0.572797
$66$	0	0
$67$	−9.56231	−1.16822	−0.584111	−	0.811674i	$-0.698557\pi$
$67$	−9.56231	−1.16822	−0.584111	+	0.811674i	$0.698557\pi$
$68$	−2.61803	−0.317483
$69$	0	0
$70$	4.85410	0.580176
$71$	5.56231	0.660124	0.330062	−	0.943959i	$-0.392930\pi$
$71$	5.56231	0.660124	0.330062	+	0.943959i	$0.392930\pi$
$72$	0	0
$73$	3.23607	0.378753	0.189377	−	0.981905i	$-0.439353\pi$
$73$	3.23607	0.378753	0.189377	+	0.981905i	$0.439353\pi$
$74$	−0.145898	−0.0169603
$75$	0	0
$76$	9.47214	1.08653
$77$	0	0
$78$	0	0
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	4.85410	0.542705
$81$	0	0
$82$	7.38197	0.815202
$83$	0.708204	0.0777355	0.0388677	−	0.999244i	$-0.487625\pi$
$83$	0.708204	0.0777355	0.0388677	+	0.999244i	$0.487625\pi$
$84$	0	0
$85$	4.23607	0.459466
$86$	3.85410	0.415599
$87$	0	0
$88$	0	0
$89$	−0.527864	−0.0559535	−0.0279767	−	0.999609i	$-0.508906\pi$
$89$	−0.527864	−0.0559535	−0.0279767	+	0.999609i	$0.508906\pi$
$90$	0	0
$91$	5.29180	0.554731
$92$	5.61803	0.585721
$93$	0	0
$94$	1.00000	0.103142
$95$	−15.3262	−1.57244
$96$	0	0
$97$	−14.0344	−1.42498	−0.712491	−	0.701681i	$-0.752433\pi$
$97$	−14.0344	−1.42498	−0.712491	+	0.701681i	$0.752433\pi$
$98$	−1.23607	−0.124862
$99$	0	0
$100$	−3.00000	−0.300000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−3.94427	−0.386768
$105$	0	0
$106$	−5.94427	−0.577359
$107$	−4.23607	−0.409516	−0.204758	−	0.978813i	$-0.565641\pi$
$107$	−4.23607	−0.409516	−0.204758	+	0.978813i	$0.565641\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	−5.56231	−0.525589
$113$	−0.708204	−0.0666222	−0.0333111	−	0.999445i	$-0.510605\pi$
$113$	−0.708204	−0.0666222	−0.0333111	+	0.999445i	$0.510605\pi$
$114$	0	0
$115$	−9.09017	−0.847663
$116$	−7.23607	−0.671852
$117$	0	0
$118$	6.38197	0.587508
$119$	−4.85410	−0.444975
$120$	0	0
$121$	0	0
$122$	4.85410	0.439470
$123$	0	0
$124$	−4.61803	−0.414712
$125$	−8.23607	−0.736656
$126$	0	0
$127$	3.70820	0.329050	0.164525	−	0.986373i	$-0.447391\pi$
$127$	3.70820	0.329050	0.164525	+	0.986373i	$0.447391\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	2.85410	0.250321
$131$	−7.14590	−0.624340	−0.312170	−	0.950026i	$-0.601056\pi$
$131$	−7.14590	−0.624340	−0.312170	+	0.950026i	$0.601056\pi$
$132$	0	0
$133$	17.5623	1.52285
$134$	5.90983	0.510532
$135$	0	0
$136$	3.61803	0.310244
$137$	−7.47214	−0.638388	−0.319194	−	0.947689i	$-0.603412\pi$
$137$	−7.47214	−0.638388	−0.319194	+	0.947689i	$0.603412\pi$
$138$	0	0
$139$	0.854102	0.0724440	0.0362220	−	0.999344i	$-0.488468\pi$
$139$	0.854102	0.0724440	0.0362220	+	0.999344i	$0.488468\pi$
$140$	12.7082	1.07404
$141$	0	0
$142$	−3.43769	−0.288485
$143$	0	0
$144$	0	0
$145$	11.7082	0.972313
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−0.381966	−0.0313974
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−13.0902	−1.06175
$153$	0	0
$154$	0	0
$155$	7.47214	0.600176
$156$	0	0
$157$	−3.70820	−0.295947	−0.147973	−	0.988991i	$-0.547275\pi$
$157$	−3.70820	−0.295947	−0.147973	+	0.988991i	$0.547275\pi$
$158$	5.85410	0.465727
$159$	0	0
$160$	−14.7082	−1.16279
$161$	10.4164	0.820928
$162$	0	0
$163$	18.2705	1.43106	0.715528	−	0.698584i	$-0.246186\pi$
$163$	18.2705	1.43106	0.715528	+	0.698584i	$0.246186\pi$
$164$	19.3262	1.50913
$165$	0	0
$166$	−0.437694	−0.0339717
$167$	10.0344	0.776488	0.388244	−	0.921557i	$-0.373082\pi$
$167$	10.0344	0.776488	0.388244	+	0.921557i	$0.373082\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	−2.61803	−0.200794
$171$	0	0
$172$	10.0902	0.769368
$173$	15.3820	1.16947	0.584735	−	0.811225i	$-0.301199\pi$
$173$	15.3820	1.16947	0.584735	+	0.811225i	$0.301199\pi$
$174$	0	0
$175$	−5.56231	−0.420471
$176$	0	0
$177$	0	0
$178$	0.326238	0.0244526
$179$	2.23607	0.167132	0.0835658	−	0.996502i	$-0.473369\pi$
$179$	2.23607	0.167132	0.0835658	+	0.996502i	$0.473369\pi$
$180$	0	0
$181$	−17.4721	−1.29869	−0.649347	−	0.760492i	$-0.724958\pi$
$181$	−17.4721	−1.29869	−0.649347	+	0.760492i	$0.724958\pi$
$182$	−3.27051	−0.242426
$183$	0	0
$184$	−7.76393	−0.572365
$185$	0.618034	0.0454388
$186$	0	0
$187$	0	0
$188$	2.61803	0.190940
$189$	0	0
$190$	9.47214	0.687181
$191$	7.47214	0.540665	0.270332	−	0.962767i	$-0.412866\pi$
$191$	7.47214	0.540665	0.270332	+	0.962767i	$0.412866\pi$
$192$	0	0
$193$	18.5623	1.33614	0.668072	−	0.744097i	$-0.267120\pi$
$193$	18.5623	1.33614	0.668072	+	0.744097i	$0.267120\pi$
$194$	8.67376	0.622740
$195$	0	0
$196$	−3.23607	−0.231148
$197$	24.3820	1.73714	0.868572	−	0.495564i	$-0.165039\pi$
$197$	24.3820	1.73714	0.868572	+	0.495564i	$0.165039\pi$
$198$	0	0
$199$	−16.7082	−1.18441	−0.592207	−	0.805786i	$-0.701743\pi$
$199$	−16.7082	−1.18441	−0.592207	+	0.805786i	$0.701743\pi$
$200$	4.14590	0.293159
$201$	0	0
$202$	1.85410	0.130454
$203$	−13.4164	−0.941647
$204$	0	0
$205$	−31.2705	−2.18403
$206$	3.70820	0.258363
$207$	0	0
$208$	−3.27051	−0.226769
$209$	0	0
$210$	0	0
$211$	22.2705	1.53317	0.766583	−	0.642146i	$-0.221956\pi$
$211$	22.2705	1.53317	0.766583	+	0.642146i	$0.221956\pi$
$212$	−15.5623	−1.06882
$213$	0	0
$214$	2.61803	0.178965
$215$	−16.3262	−1.11344
$216$	0	0
$217$	−8.56231	−0.581247
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.85410	−0.191988
$222$	0	0
$223$	0.708204	0.0474248	0.0237124	−	0.999719i	$-0.492451\pi$
$223$	0.708204	0.0474248	0.0237124	+	0.999719i	$0.492451\pi$
$224$	16.8541	1.12611
$225$	0	0
$226$	0.437694	0.0291150
$227$	−24.8885	−1.65191	−0.825955	−	0.563736i	$-0.809364\pi$
$227$	−24.8885	−1.65191	−0.825955	+	0.563736i	$0.809364\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	5.61803	0.370442
$231$	0	0
$232$	10.0000	0.656532
$233$	24.3262	1.59366	0.796832	−	0.604200i	$-0.206507\pi$
$233$	24.3262	1.59366	0.796832	+	0.604200i	$0.206507\pi$
$234$	0	0
$235$	−4.23607	−0.276331
$236$	16.7082	1.08761
$237$	0	0
$238$	3.00000	0.194461
$239$	2.56231	0.165742	0.0828709	−	0.996560i	$-0.473591\pi$
$239$	2.56231	0.165742	0.0828709	+	0.996560i	$0.473591\pi$
$240$	0	0
$241$	23.1246	1.48959	0.744794	−	0.667295i	$-0.232548\pi$
$241$	23.1246	1.48959	0.744794	+	0.667295i	$0.232548\pi$
$242$	0	0
$243$	0	0
$244$	12.7082	0.813559
$245$	5.23607	0.334520
$246$	0	0
$247$	10.3262	0.657043
$248$	6.38197	0.405255
$249$	0	0
$250$	5.09017	0.321931
$251$	7.79837	0.492229	0.246114	−	0.969241i	$-0.420846\pi$
$251$	7.79837	0.492229	0.246114	+	0.969241i	$0.420846\pi$
$252$	0	0
$253$	0	0
$254$	−2.29180	−0.143800
$255$	0	0
$256$	−6.56231	−0.410144
$257$	11.6738	0.728189	0.364095	−	0.931362i	$-0.381378\pi$
$257$	11.6738	0.728189	0.364095	+	0.931362i	$0.381378\pi$
$258$	0	0
$259$	−0.708204	−0.0440057
$260$	7.47214	0.463402
$261$	0	0
$262$	4.41641	0.272847
$263$	−16.3262	−1.00672	−0.503359	−	0.864077i	$-0.667903\pi$
$263$	−16.3262	−1.00672	−0.503359	+	0.864077i	$0.667903\pi$
$264$	0	0
$265$	25.1803	1.54682
$266$	−10.8541	−0.665508
$267$	0	0
$268$	15.4721	0.945111
$269$	−15.5279	−0.946751	−0.473375	−	0.880861i	$-0.656965\pi$
$269$	−15.5279	−0.946751	−0.473375	+	0.880861i	$0.656965\pi$
$270$	0	0
$271$	27.2705	1.65657	0.828283	−	0.560310i	$-0.189318\pi$
$271$	27.2705	1.65657	0.828283	+	0.560310i	$0.189318\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	4.61803	0.278986
$275$	0	0
$276$	0	0
$277$	−30.5623	−1.83631	−0.918155	−	0.396220i	$-0.870322\pi$
$277$	−30.5623	−1.83631	−0.918155	+	0.396220i	$0.870322\pi$
$278$	−0.527864	−0.0316592
$279$	0	0
$280$	−17.5623	−1.04955
$281$	−0.763932	−0.0455724	−0.0227862	−	0.999740i	$-0.507254\pi$
$281$	−0.763932	−0.0455724	−0.0227862	+	0.999740i	$0.507254\pi$
$282$	0	0
$283$	−0.180340	−0.0107201	−0.00536005	−	0.999986i	$-0.501706\pi$
$283$	−0.180340	−0.0107201	−0.00536005	+	0.999986i	$0.501706\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	0	0
$287$	35.8328	2.11514
$288$	0	0
$289$	−14.3820	−0.845998
$290$	−7.23607	−0.424917
$291$	0	0
$292$	−5.23607	−0.306418
$293$	0.0557281	0.00325567	0.00162783	−	0.999999i	$-0.499482\pi$
$293$	0.0557281	0.00325567	0.00162783	+	0.999999i	$0.499482\pi$
$294$	0	0
$295$	−27.0344	−1.57401
$296$	0.527864	0.0306815
$297$	0	0
$298$	−9.27051	−0.537026
$299$	6.12461	0.354195
$300$	0	0
$301$	18.7082	1.07832
$302$	1.23607	0.0711277
$303$	0	0
$304$	−10.8541	−0.622525
$305$	−20.5623	−1.17739
$306$	0	0
$307$	−0.562306	−0.0320925	−0.0160462	−	0.999871i	$-0.505108\pi$
$307$	−0.562306	−0.0320925	−0.0160462	+	0.999871i	$0.505108\pi$
$308$	0	0
$309$	0	0
$310$	−4.61803	−0.262287
$311$	−2.52786	−0.143342	−0.0716710	−	0.997428i	$-0.522833\pi$
$311$	−2.52786	−0.143342	−0.0716710	+	0.997428i	$0.522833\pi$
$312$	0	0
$313$	25.8328	1.46016	0.730079	−	0.683363i	$-0.239483\pi$
$313$	25.8328	1.46016	0.730079	+	0.683363i	$0.239483\pi$
$314$	2.29180	0.129334
$315$	0	0
$316$	15.3262	0.862168
$317$	19.3607	1.08740	0.543702	−	0.839278i	$-0.317022\pi$
$317$	19.3607	1.08740	0.543702	+	0.839278i	$0.317022\pi$
$318$	0	0
$319$	0	0
$320$	−0.618034	−0.0345492
$321$	0	0
$322$	−6.43769	−0.358759
$323$	−9.47214	−0.527044
$324$	0	0
$325$	−3.27051	−0.181415
$326$	−11.2918	−0.625395
$327$	0	0
$328$	−26.7082	−1.47471
$329$	4.85410	0.267615
$330$	0	0
$331$	26.5967	1.46189	0.730945	−	0.682437i	$-0.239080\pi$
$331$	26.5967	1.46189	0.730945	+	0.682437i	$0.239080\pi$
$332$	−1.14590	−0.0628893
$333$	0	0
$334$	−6.20163	−0.339338
$335$	−25.0344	−1.36778
$336$	0	0
$337$	0.291796	0.0158951	0.00794757	−	0.999968i	$-0.497470\pi$
$337$	0.291796	0.0158951	0.00794757	+	0.999968i	$0.497470\pi$
$338$	6.11146	0.332419
$339$	0	0
$340$	−6.85410	−0.371716
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	−13.9443	−0.751825
$345$	0	0
$346$	−9.50658	−0.511077
$347$	−20.9443	−1.12435	−0.562174	−	0.827019i	$-0.690035\pi$
$347$	−20.9443	−1.12435	−0.562174	+	0.827019i	$0.690035\pi$
$348$	0	0
$349$	10.1246	0.541958	0.270979	−	0.962585i	$-0.412653\pi$
$349$	10.1246	0.541958	0.270979	+	0.962585i	$0.412653\pi$
$350$	3.43769	0.183752
$351$	0	0
$352$	0	0
$353$	10.4721	0.557376	0.278688	−	0.960382i	$-0.410101\pi$
$353$	10.4721	0.557376	0.278688	+	0.960382i	$0.410101\pi$
$354$	0	0
$355$	14.5623	0.772887
$356$	0.854102	0.0452673
$357$	0	0
$358$	−1.38197	−0.0730392
$359$	−12.7639	−0.673655	−0.336827	−	0.941566i	$-0.609354\pi$
$359$	−12.7639	−0.673655	−0.336827	+	0.941566i	$0.609354\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	10.7984	0.567550
$363$	0	0
$364$	−8.56231	−0.448787
$365$	8.47214	0.443452
$366$	0	0
$367$	5.56231	0.290350	0.145175	−	0.989406i	$-0.453625\pi$
$367$	5.56231	0.290350	0.145175	+	0.989406i	$0.453625\pi$
$368$	−6.43769	−0.335588
$369$	0	0
$370$	−0.381966	−0.0198575
$371$	−28.8541	−1.49803
$372$	0	0
$373$	4.41641	0.228673	0.114336	−	0.993442i	$-0.463526\pi$
$373$	4.41641	0.228673	0.114336	+	0.993442i	$0.463526\pi$
$374$	0	0
$375$	0	0
$376$	−3.61803	−0.186586
$377$	−7.88854	−0.406281
$378$	0	0
$379$	1.58359	0.0813437	0.0406718	−	0.999173i	$-0.487050\pi$
$379$	1.58359	0.0813437	0.0406718	+	0.999173i	$0.487050\pi$
$380$	24.7984	1.27213
$381$	0	0
$382$	−4.61803	−0.236279
$383$	−26.8885	−1.37394	−0.686970	−	0.726686i	$-0.741060\pi$
$383$	−26.8885	−1.37394	−0.686970	+	0.726686i	$0.741060\pi$
$384$	0	0
$385$	0	0
$386$	−11.4721	−0.583916
$387$	0	0
$388$	22.7082	1.15283
$389$	−24.2705	−1.23056	−0.615282	−	0.788307i	$-0.710958\pi$
$389$	−24.2705	−1.23056	−0.615282	+	0.788307i	$0.710958\pi$
$390$	0	0
$391$	−5.61803	−0.284116
$392$	4.47214	0.225877
$393$	0	0
$394$	−15.0689	−0.759159
$395$	−24.7984	−1.24774
$396$	0	0
$397$	−38.7082	−1.94271	−0.971355	−	0.237635i	$-0.923628\pi$
$397$	−38.7082	−1.94271	−0.971355	+	0.237635i	$0.923628\pi$
$398$	10.3262	0.517608
$399$	0	0
$400$	3.43769	0.171885
$401$	26.0902	1.30288	0.651440	−	0.758700i	$-0.274165\pi$
$401$	26.0902	1.30288	0.651440	+	0.758700i	$0.274165\pi$
$402$	0	0
$403$	−5.03444	−0.250783
$404$	4.85410	0.241501
$405$	0	0
$406$	8.29180	0.411515
$407$	0	0
$408$	0	0
$409$	11.0557	0.546671	0.273335	−	0.961919i	$-0.411873\pi$
$409$	11.0557	0.546671	0.273335	+	0.961919i	$0.411873\pi$
$410$	19.3262	0.954455
$411$	0	0
$412$	9.70820	0.478289
$413$	30.9787	1.52436
$414$	0	0
$415$	1.85410	0.0910143
$416$	9.90983	0.485869
$417$	0	0
$418$	0	0
$419$	−16.5066	−0.806399	−0.403200	−	0.915112i	$-0.632102\pi$
$419$	−16.5066	−0.806399	−0.403200	+	0.915112i	$0.632102\pi$
$420$	0	0
$421$	−37.2705	−1.81645	−0.908227	−	0.418478i	$-0.862564\pi$
$421$	−37.2705	−1.81645	−0.908227	+	0.418478i	$0.862564\pi$
$422$	−13.7639	−0.670018
$423$	0	0
$424$	21.5066	1.04445
$425$	3.00000	0.145521
$426$	0	0
$427$	23.5623	1.14026
$428$	6.85410	0.331306
$429$	0	0
$430$	10.0902	0.486591
$431$	−39.5066	−1.90296	−0.951482	−	0.307703i	$-0.900440\pi$
$431$	−39.5066	−1.90296	−0.951482	+	0.307703i	$0.900440\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	5.29180	0.254014
$435$	0	0
$436$	0	0
$437$	20.3262	0.972336
$438$	0	0
$439$	−3.29180	−0.157109	−0.0785544	−	0.996910i	$-0.525030\pi$
$439$	−3.29180	−0.157109	−0.0785544	+	0.996910i	$0.525030\pi$
$440$	0	0
$441$	0	0
$442$	1.76393	0.0839017
$443$	41.1246	1.95389	0.976945	−	0.213493i	$-0.0684840\pi$
$443$	41.1246	1.95389	0.976945	+	0.213493i	$0.0684840\pi$
$444$	0	0
$445$	−1.38197	−0.0655115
$446$	−0.437694	−0.0207254
$447$	0	0
$448$	0.708204	0.0334595
$449$	24.4721	1.15491	0.577456	−	0.816422i	$-0.304046\pi$
$449$	24.4721	1.15491	0.577456	+	0.816422i	$0.304046\pi$
$450$	0	0
$451$	0	0
$452$	1.14590	0.0538985
$453$	0	0
$454$	15.3820	0.721911
$455$	13.8541	0.649490
$456$	0	0
$457$	−8.20163	−0.383656	−0.191828	−	0.981429i	$-0.561442\pi$
$457$	−8.20163	−0.383656	−0.191828	+	0.981429i	$0.561442\pi$
$458$	−6.18034	−0.288788
$459$	0	0
$460$	14.7082	0.685774
$461$	−21.0902	−0.982267	−0.491134	−	0.871084i	$-0.663417\pi$
$461$	−21.0902	−0.982267	−0.491134	+	0.871084i	$0.663417\pi$
$462$	0	0
$463$	−15.7984	−0.734213	−0.367106	−	0.930179i	$-0.619651\pi$
$463$	−15.7984	−0.734213	−0.367106	+	0.930179i	$0.619651\pi$
$464$	8.29180	0.384937
$465$	0	0
$466$	−15.0344	−0.696457
$467$	9.76393	0.451821	0.225910	−	0.974148i	$-0.427464\pi$
$467$	9.76393	0.451821	0.225910	+	0.974148i	$0.427464\pi$
$468$	0	0
$469$	28.6869	1.32464
$470$	2.61803	0.120761
$471$	0	0
$472$	−23.0902	−1.06281
$473$	0	0
$474$	0	0
$475$	−10.8541	−0.498020
$476$	7.85410	0.359992
$477$	0	0
$478$	−1.58359	−0.0724318
$479$	−28.0902	−1.28347	−0.641736	−	0.766925i	$-0.721786\pi$
$479$	−28.0902	−1.28347	−0.641736	+	0.766925i	$0.721786\pi$
$480$	0	0
$481$	−0.416408	−0.0189866
$482$	−14.2918	−0.650973
$483$	0	0
$484$	0	0
$485$	−36.7426	−1.66840
$486$	0	0
$487$	16.8197	0.762172	0.381086	−	0.924540i	$-0.375550\pi$
$487$	16.8197	0.762172	0.381086	+	0.924540i	$0.375550\pi$
$488$	−17.5623	−0.795008
$489$	0	0
$490$	−3.23607	−0.146191
$491$	25.2148	1.13793	0.568964	−	0.822363i	$-0.307345\pi$
$491$	25.2148	1.13793	0.568964	+	0.822363i	$0.307345\pi$
$492$	0	0
$493$	7.23607	0.325896
$494$	−6.38197	−0.287138
$495$	0	0
$496$	5.29180	0.237609
$497$	−16.6869	−0.748511
$498$	0	0
$499$	−17.5623	−0.786197	−0.393098	−	0.919496i	$-0.628597\pi$
$499$	−17.5623	−0.786197	−0.393098	+	0.919496i	$0.628597\pi$
$500$	13.3262	0.595967
$501$	0	0
$502$	−4.81966	−0.215112
$503$	28.0689	1.25153	0.625765	−	0.780012i	$-0.284787\pi$
$503$	28.0689	1.25153	0.625765	+	0.780012i	$0.284787\pi$
$504$	0	0
$505$	−7.85410	−0.349503
$506$	0	0
$507$	0	0
$508$	−6.00000	−0.266207
$509$	−23.6180	−1.04685	−0.523425	−	0.852071i	$-0.675346\pi$
$509$	−23.6180	−1.04685	−0.523425	+	0.852071i	$0.675346\pi$
$510$	0	0
$511$	−9.70820	−0.429466
$512$	−18.7082	−0.826794
$513$	0	0
$514$	−7.21478	−0.318230
$515$	−15.7082	−0.692186
$516$	0	0
$517$	0	0
$518$	0.437694	0.0192312
$519$	0	0
$520$	−10.3262	−0.452835
$521$	14.8328	0.649837	0.324919	−	0.945742i	$-0.394663\pi$
$521$	14.8328	0.649837	0.324919	+	0.945742i	$0.394663\pi$
$522$	0	0
$523$	11.9787	0.523793	0.261896	−	0.965096i	$-0.415652\pi$
$523$	11.9787	0.523793	0.261896	+	0.965096i	$0.415652\pi$
$524$	11.5623	0.505102
$525$	0	0
$526$	10.0902	0.439952
$527$	4.61803	0.201165
$528$	0	0
$529$	−10.9443	−0.475838
$530$	−15.5623	−0.675983
$531$	0	0
$532$	−28.4164	−1.23201
$533$	21.0689	0.912595
$534$	0	0
$535$	−11.0902	−0.479470
$536$	−21.3820	−0.923560
$537$	0	0
$538$	9.59675	0.413745
$539$	0	0
$540$	0	0
$541$	−19.5623	−0.841049	−0.420525	−	0.907281i	$-0.638154\pi$
$541$	−19.5623	−0.841049	−0.420525	+	0.907281i	$0.638154\pi$
$542$	−16.8541	−0.723946
$543$	0	0
$544$	−9.09017	−0.389738
$545$	0	0
$546$	0	0
$547$	−21.6180	−0.924320	−0.462160	−	0.886796i	$-0.652925\pi$
$547$	−21.6180	−0.924320	−0.462160	+	0.886796i	$0.652925\pi$
$548$	12.0902	0.516466
$549$	0	0
$550$	0	0
$551$	−26.1803	−1.11532
$552$	0	0
$553$	28.4164	1.20839
$554$	18.8885	0.802497
$555$	0	0
$556$	−1.38197	−0.0586084
$557$	14.9098	0.631750	0.315875	−	0.948801i	$-0.397702\pi$
$557$	14.9098	0.631750	0.315875	+	0.948801i	$0.397702\pi$
$558$	0	0
$559$	11.0000	0.465250
$560$	−14.5623	−0.615370
$561$	0	0
$562$	0.472136	0.0199159
$563$	−8.88854	−0.374607	−0.187304	−	0.982302i	$-0.559975\pi$
$563$	−8.88854	−0.374607	−0.187304	+	0.982302i	$0.559975\pi$
$564$	0	0
$565$	−1.85410	−0.0780027
$566$	0.111456	0.00468485
$567$	0	0
$568$	12.4377	0.521874
$569$	−24.0689	−1.00902	−0.504510	−	0.863406i	$-0.668327\pi$
$569$	−24.0689	−1.00902	−0.504510	+	0.863406i	$0.668327\pi$
$570$	0	0
$571$	−34.6869	−1.45160	−0.725801	−	0.687905i	$-0.758531\pi$
$571$	−34.6869	−1.45160	−0.725801	+	0.687905i	$0.758531\pi$
$572$	0	0
$573$	0	0
$574$	−22.1459	−0.924352
$575$	−6.43769	−0.268470
$576$	0	0
$577$	10.7639	0.448108	0.224054	−	0.974577i	$-0.428071\pi$
$577$	10.7639	0.448108	0.224054	+	0.974577i	$0.428071\pi$
$578$	8.88854	0.369715
$579$	0	0
$580$	−18.9443	−0.786618
$581$	−2.12461	−0.0881437
$582$	0	0
$583$	0	0
$584$	7.23607	0.299431
$585$	0	0
$586$	−0.0344419	−0.00142278
$587$	38.3050	1.58101	0.790507	−	0.612453i	$-0.209817\pi$
$587$	38.3050	1.58101	0.790507	+	0.612453i	$0.209817\pi$
$588$	0	0
$589$	−16.7082	−0.688450
$590$	16.7082	0.687866
$591$	0	0
$592$	0.437694	0.0179891
$593$	22.2148	0.912252	0.456126	−	0.889915i	$-0.349237\pi$
$593$	22.2148	0.912252	0.456126	+	0.889915i	$0.349237\pi$
$594$	0	0
$595$	−12.7082	−0.520986
$596$	−24.2705	−0.994159
$597$	0	0
$598$	−3.78522	−0.154789
$599$	8.29180	0.338794	0.169397	−	0.985548i	$-0.445818\pi$
$599$	8.29180	0.338794	0.169397	+	0.985548i	$0.445818\pi$
$600$	0	0
$601$	−33.8328	−1.38007	−0.690035	−	0.723776i	$-0.742405\pi$
$601$	−33.8328	−1.38007	−0.690035	+	0.723776i	$0.742405\pi$
$602$	−11.5623	−0.471244
$603$	0	0
$604$	3.23607	0.131674
$605$	0	0
$606$	0	0
$607$	−13.3262	−0.540895	−0.270448	−	0.962735i	$-0.587172\pi$
$607$	−13.3262	−0.540895	−0.270448	+	0.962735i	$0.587172\pi$
$608$	32.8885	1.33381
$609$	0	0
$610$	12.7082	0.514540
$611$	2.85410	0.115465
$612$	0	0
$613$	−34.6525	−1.39960	−0.699800	−	0.714339i	$-0.746728\pi$
$613$	−34.6525	−1.39960	−0.699800	+	0.714339i	$0.746728\pi$
$614$	0.347524	0.0140249
$615$	0	0
$616$	0	0
$617$	−19.5836	−0.788406	−0.394203	−	0.919023i	$-0.628979\pi$
$617$	−19.5836	−0.788406	−0.394203	+	0.919023i	$0.628979\pi$
$618$	0	0
$619$	−8.81966	−0.354492	−0.177246	−	0.984167i	$-0.556719\pi$
$619$	−8.81966	−0.354492	−0.177246	+	0.984167i	$0.556719\pi$
$620$	−12.0902	−0.485553
$621$	0	0
$622$	1.56231	0.0626428
$623$	1.58359	0.0634453
$624$	0	0
$625$	−30.8328	−1.23331
$626$	−15.9656	−0.638112
$627$	0	0
$628$	6.00000	0.239426
$629$	0.381966	0.0152300
$630$	0	0
$631$	46.2705	1.84200	0.921000	−	0.389563i	$-0.127374\pi$
$631$	46.2705	1.84200	0.921000	+	0.389563i	$0.127374\pi$
$632$	−21.1803	−0.842509
$633$	0	0
$634$	−11.9656	−0.475213
$635$	9.70820	0.385258
$636$	0	0
$637$	−3.52786	−0.139779
$638$	0	0
$639$	0	0
$640$	29.7984	1.17788
$641$	−35.7426	−1.41175	−0.705875	−	0.708337i	$-0.749446\pi$
$641$	−35.7426	−1.41175	−0.705875	+	0.708337i	$0.749446\pi$
$642$	0	0
$643$	−38.5623	−1.52075	−0.760374	−	0.649485i	$-0.774985\pi$
$643$	−38.5623	−1.52075	−0.760374	+	0.649485i	$0.774985\pi$
$644$	−16.8541	−0.664145
$645$	0	0
$646$	5.85410	0.230327
$647$	−27.7984	−1.09287	−0.546433	−	0.837503i	$-0.684015\pi$
$647$	−27.7984	−1.09287	−0.546433	+	0.837503i	$0.684015\pi$
$648$	0	0
$649$	0	0
$650$	2.02129	0.0792814
$651$	0	0
$652$	−29.5623	−1.15775
$653$	−51.0344	−1.99713	−0.998566	−	0.0535342i	$-0.982951\pi$
$653$	−51.0344	−1.99713	−0.998566	+	0.0535342i	$0.982951\pi$
$654$	0	0
$655$	−18.7082	−0.730990
$656$	−22.1459	−0.864652
$657$	0	0
$658$	−3.00000	−0.116952
$659$	−10.6525	−0.414962	−0.207481	−	0.978239i	$-0.566526\pi$
$659$	−10.6525	−0.414962	−0.207481	+	0.978239i	$0.566526\pi$
$660$	0	0
$661$	−9.90983	−0.385448	−0.192724	−	0.981253i	$-0.561732\pi$
$661$	−9.90983	−0.385448	−0.192724	+	0.981253i	$0.561732\pi$
$662$	−16.4377	−0.638869
$663$	0	0
$664$	1.58359	0.0614553
$665$	45.9787	1.78298
$666$	0	0
$667$	−15.5279	−0.601241
$668$	−16.2361	−0.628192
$669$	0	0
$670$	15.4721	0.597741
$671$	0	0
$672$	0	0
$673$	−12.4164	−0.478617	−0.239309	−	0.970944i	$-0.576921\pi$
$673$	−12.4164	−0.478617	−0.239309	+	0.970944i	$0.576921\pi$
$674$	−0.180340	−0.00694643
$675$	0	0
$676$	16.0000	0.615385
$677$	13.5279	0.519918	0.259959	−	0.965620i	$-0.416291\pi$
$677$	13.5279	0.519918	0.259959	+	0.965620i	$0.416291\pi$
$678$	0	0
$679$	42.1033	1.61578
$680$	9.47214	0.363240
$681$	0	0
$682$	0	0
$683$	3.11146	0.119057	0.0595283	−	0.998227i	$-0.481040\pi$
$683$	3.11146	0.119057	0.0595283	+	0.998227i	$0.481040\pi$
$684$	0	0
$685$	−19.5623	−0.747437
$686$	−9.27051	−0.353950
$687$	0	0
$688$	−11.5623	−0.440809
$689$	−16.9656	−0.646336
$690$	0	0
$691$	−26.2918	−1.00019	−0.500094	−	0.865971i	$-0.666701\pi$
$691$	−26.2918	−1.00019	−0.500094	+	0.865971i	$0.666701\pi$
$692$	−24.8885	−0.946120
$693$	0	0
$694$	12.9443	0.491358
$695$	2.23607	0.0848189
$696$	0	0
$697$	−19.3262	−0.732033
$698$	−6.25735	−0.236844
$699$	0	0
$700$	9.00000	0.340168
$701$	−10.6869	−0.403639	−0.201820	−	0.979423i	$-0.564685\pi$
$701$	−10.6869	−0.403639	−0.201820	+	0.979423i	$0.564685\pi$
$702$	0	0
$703$	−1.38197	−0.0521218
$704$	0	0
$705$	0	0
$706$	−6.47214	−0.243582
$707$	9.00000	0.338480
$708$	0	0
$709$	48.7426	1.83057	0.915284	−	0.402809i	$-0.131966\pi$
$709$	48.7426	1.83057	0.915284	+	0.402809i	$0.131966\pi$
$710$	−9.00000	−0.337764
$711$	0	0
$712$	−1.18034	−0.0442351
$713$	−9.90983	−0.371126
$714$	0	0
$715$	0	0
$716$	−3.61803	−0.135212
$717$	0	0
$718$	7.88854	0.294398
$719$	1.58359	0.0590580	0.0295290	−	0.999564i	$-0.490599\pi$
$719$	1.58359	0.0590580	0.0295290	+	0.999564i	$0.490599\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	−9.43769	−0.351235
$723$	0	0
$724$	28.2705	1.05067
$725$	8.29180	0.307950
$726$	0	0
$727$	38.8541	1.44102	0.720509	−	0.693445i	$-0.243908\pi$
$727$	38.8541	1.44102	0.720509	+	0.693445i	$0.243908\pi$
$728$	11.8328	0.438553
$729$	0	0
$730$	−5.23607	−0.193796
$731$	−10.0902	−0.373198
$732$	0	0
$733$	37.7082	1.39278	0.696392	−	0.717661i	$-0.254787\pi$
$733$	37.7082	1.39278	0.696392	+	0.717661i	$0.254787\pi$
$734$	−3.43769	−0.126888
$735$	0	0
$736$	19.5066	0.719022
$737$	0	0
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	17.8328	0.654663
$743$	35.1803	1.29064	0.645321	−	0.763912i	$-0.276724\pi$
$743$	35.1803	1.29064	0.645321	+	0.763912i	$0.276724\pi$
$744$	0	0
$745$	39.2705	1.43876
$746$	−2.72949	−0.0999337
$747$	0	0
$748$	0	0
$749$	12.7082	0.464348
$750$	0	0
$751$	11.9230	0.435076	0.217538	−	0.976052i	$-0.430197\pi$
$751$	11.9230	0.435076	0.217538	+	0.976052i	$0.430197\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	4.87539	0.177551
$755$	−5.23607	−0.190560
$756$	0	0
$757$	1.94427	0.0706658	0.0353329	−	0.999376i	$-0.488751\pi$
$757$	1.94427	0.0706658	0.0353329	+	0.999376i	$0.488751\pi$
$758$	−0.978714	−0.0355485
$759$	0	0
$760$	−34.2705	−1.24312
$761$	−30.8885	−1.11971	−0.559854	−	0.828591i	$-0.689143\pi$
$761$	−30.8885	−1.11971	−0.559854	+	0.828591i	$0.689143\pi$
$762$	0	0
$763$	0	0
$764$	−12.0902	−0.437407
$765$	0	0
$766$	16.6180	0.600434
$767$	18.2148	0.657698
$768$	0	0
$769$	−12.6869	−0.457502	−0.228751	−	0.973485i	$-0.573464\pi$
$769$	−12.6869	−0.457502	−0.228751	+	0.973485i	$0.573464\pi$
$770$	0	0
$771$	0	0
$772$	−30.0344	−1.08096
$773$	31.2492	1.12396	0.561978	−	0.827152i	$-0.310040\pi$
$773$	31.2492	1.12396	0.561978	+	0.827152i	$0.310040\pi$
$774$	0	0
$775$	5.29180	0.190087
$776$	−31.3820	−1.12655
$777$	0	0
$778$	15.0000	0.537776
$779$	69.9230	2.50525
$780$	0	0
$781$	0	0
$782$	3.47214	0.124163
$783$	0	0
$784$	3.70820	0.132436
$785$	−9.70820	−0.346501
$786$	0	0
$787$	10.2918	0.366863	0.183431	−	0.983033i	$-0.441279\pi$
$787$	10.2918	0.366863	0.183431	+	0.983033i	$0.441279\pi$
$788$	−39.4508	−1.40538
$789$	0	0
$790$	15.3262	0.545283
$791$	2.12461	0.0755425
$792$	0	0
$793$	13.8541	0.491974
$794$	23.9230	0.848995
$795$	0	0
$796$	27.0344	0.958210
$797$	−10.7639	−0.381278	−0.190639	−	0.981660i	$-0.561056\pi$
$797$	−10.7639	−0.381278	−0.190639	+	0.981660i	$0.561056\pi$
$798$	0	0
$799$	−2.61803	−0.0926194
$800$	−10.4164	−0.368276
$801$	0	0
$802$	−16.1246	−0.569380
$803$	0	0
$804$	0	0
$805$	27.2705	0.961159
$806$	3.11146	0.109596
$807$	0	0
$808$	−6.70820	−0.235994
$809$	9.67376	0.340111	0.170056	−	0.985434i	$-0.445605\pi$
$809$	9.67376	0.340111	0.170056	+	0.985434i	$0.445605\pi$
$810$	0	0
$811$	−3.25735	−0.114381	−0.0571906	−	0.998363i	$-0.518214\pi$
$811$	−3.25735	−0.114381	−0.0571906	+	0.998363i	$0.518214\pi$
$812$	21.7082	0.761809
$813$	0	0
$814$	0	0
$815$	47.8328	1.67551
$816$	0	0
$817$	36.5066	1.27720
$818$	−6.83282	−0.238904
$819$	0	0
$820$	50.5967	1.76692
$821$	40.4164	1.41054	0.705271	−	0.708938i	$-0.250825\pi$
$821$	40.4164	1.41054	0.705271	+	0.708938i	$0.250825\pi$
$822$	0	0
$823$	−35.4721	−1.23648	−0.618240	−	0.785989i	$-0.712154\pi$
$823$	−35.4721	−1.23648	−0.618240	+	0.785989i	$0.712154\pi$
$824$	−13.4164	−0.467383
$825$	0	0
$826$	−19.1459	−0.666171
$827$	53.1246	1.84732	0.923662	−	0.383208i	$-0.125181\pi$
$827$	53.1246	1.84732	0.923662	+	0.383208i	$0.125181\pi$
$828$	0	0
$829$	17.6869	0.614292	0.307146	−	0.951662i	$-0.400626\pi$
$829$	17.6869	0.614292	0.307146	+	0.951662i	$0.400626\pi$
$830$	−1.14590	−0.0397747
$831$	0	0
$832$	0.416408	0.0144363
$833$	3.23607	0.112123
$834$	0	0
$835$	26.2705	0.909128
$836$	0	0
$837$	0	0
$838$	10.2016	0.352409
$839$	36.7082	1.26731	0.633654	−	0.773617i	$-0.281554\pi$
$839$	36.7082	1.26731	0.633654	+	0.773617i	$0.281554\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	23.0344	0.793819
$843$	0	0
$844$	−36.0344	−1.24036
$845$	−25.8885	−0.890593
$846$	0	0
$847$	0	0
$848$	17.8328	0.612381
$849$	0	0
$850$	−1.85410	−0.0635952
$851$	−0.819660	−0.0280976
$852$	0	0
$853$	9.94427	0.340485	0.170243	−	0.985402i	$-0.445545\pi$
$853$	9.94427	0.340485	0.170243	+	0.985402i	$0.445545\pi$
$854$	−14.5623	−0.498312
$855$	0	0
$856$	−9.47214	−0.323751
$857$	47.7214	1.63013	0.815065	−	0.579369i	$-0.196701\pi$
$857$	47.7214	1.63013	0.815065	+	0.579369i	$0.196701\pi$
$858$	0	0
$859$	7.11146	0.242640	0.121320	−	0.992613i	$-0.461287\pi$
$859$	7.11146	0.242640	0.121320	+	0.992613i	$0.461287\pi$
$860$	26.4164	0.900792
$861$	0	0
$862$	24.4164	0.831626
$863$	−11.8885	−0.404691	−0.202345	−	0.979314i	$-0.564856\pi$
$863$	−11.8885	−0.404691	−0.202345	+	0.979314i	$0.564856\pi$
$864$	0	0
$865$	40.2705	1.36924
$866$	3.70820	0.126010
$867$	0	0
$868$	13.8541	0.470239
$869$	0	0
$870$	0	0
$871$	16.8673	0.571525
$872$	0	0
$873$	0	0
$874$	−12.5623	−0.424926
$875$	24.7082	0.835290
$876$	0	0
$877$	−6.41641	−0.216667	−0.108333	−	0.994115i	$-0.534551\pi$
$877$	−6.41641	−0.216667	−0.108333	+	0.994115i	$0.534551\pi$
$878$	2.03444	0.0686591
$879$	0	0
$880$	0	0
$881$	−13.9098	−0.468634	−0.234317	−	0.972160i	$-0.575285\pi$
$881$	−13.9098	−0.468634	−0.234317	+	0.972160i	$0.575285\pi$
$882$	0	0
$883$	10.5836	0.356166	0.178083	−	0.984015i	$-0.443010\pi$
$883$	10.5836	0.356166	0.178083	+	0.984015i	$0.443010\pi$
$884$	4.61803	0.155321
$885$	0	0
$886$	−25.4164	−0.853881
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	−11.1246	−0.373108
$890$	0.854102	0.0286296
$891$	0	0
$892$	−1.14590	−0.0383675
$893$	9.47214	0.316973
$894$	0	0
$895$	5.85410	0.195681
$896$	−34.1459	−1.14073
$897$	0	0
$898$	−15.1246	−0.504715
$899$	12.7639	0.425701
$900$	0	0
$901$	15.5623	0.518456
$902$	0	0
$903$	0	0
$904$	−1.58359	−0.0526695
$905$	−45.7426	−1.52054
$906$	0	0
$907$	−42.9787	−1.42708	−0.713542	−	0.700612i	$-0.752910\pi$
$907$	−42.9787	−1.42708	−0.713542	+	0.700612i	$0.752910\pi$
$908$	40.2705	1.33642
$909$	0	0
$910$	−8.56231	−0.283838
$911$	−18.0557	−0.598213	−0.299106	−	0.954220i	$-0.596689\pi$
$911$	−18.0557	−0.598213	−0.299106	+	0.954220i	$0.596689\pi$
$912$	0	0
$913$	0	0
$914$	5.06888	0.167664
$915$	0	0
$916$	−16.1803	−0.534613
$917$	21.4377	0.707935
$918$	0	0
$919$	−46.9574	−1.54898	−0.774491	−	0.632585i	$-0.781994\pi$
$919$	−46.9574	−1.54898	−0.774491	+	0.632585i	$0.781994\pi$
$920$	−20.3262	−0.670136
$921$	0	0
$922$	13.0344	0.429266
$923$	−9.81153	−0.322950
$924$	0	0
$925$	0.437694	0.0143913
$926$	9.76393	0.320863
$927$	0	0
$928$	−25.1246	−0.824756
$929$	32.8885	1.07904	0.539519	−	0.841973i	$-0.318606\pi$
$929$	32.8885	1.07904	0.539519	+	0.841973i	$0.318606\pi$
$930$	0	0
$931$	−11.7082	−0.383721
$932$	−39.3607	−1.28930
$933$	0	0
$934$	−6.03444	−0.197453
$935$	0	0
$936$	0	0
$937$	16.5967	0.542192	0.271096	−	0.962552i	$-0.412614\pi$
$937$	16.5967	0.542192	0.271096	+	0.962552i	$0.412614\pi$
$938$	−17.7295	−0.578888
$939$	0	0
$940$	6.85410	0.223556
$941$	−44.6312	−1.45494	−0.727468	−	0.686142i	$-0.759303\pi$
$941$	−44.6312	−1.45494	−0.727468	+	0.686142i	$0.759303\pi$
$942$	0	0
$943$	41.4721	1.35052
$944$	−19.1459	−0.623146
$945$	0	0
$946$	0	0
$947$	−18.3262	−0.595523	−0.297761	−	0.954640i	$-0.596240\pi$
$947$	−18.3262	−0.595523	−0.297761	+	0.954640i	$0.596240\pi$
$948$	0	0
$949$	−5.70820	−0.185296
$950$	6.70820	0.217643
$951$	0	0
$952$	−10.8541	−0.351783
$953$	37.8197	1.22510	0.612549	−	0.790432i	$-0.290144\pi$
$953$	37.8197	1.22510	0.612549	+	0.790432i	$0.290144\pi$
$954$	0	0
$955$	19.5623	0.633021
$956$	−4.14590	−0.134088
$957$	0	0
$958$	17.3607	0.560898
$959$	22.4164	0.723864
$960$	0	0
$961$	−22.8541	−0.737229
$962$	0.257354	0.00829743
$963$	0	0
$964$	−37.4164	−1.20510
$965$	48.5967	1.56438
$966$	0	0
$967$	34.6869	1.11546	0.557728	−	0.830024i	$-0.311673\pi$
$967$	34.6869	1.11546	0.557728	+	0.830024i	$0.311673\pi$
$968$	0	0
$969$	0	0
$970$	22.7082	0.729116
$971$	−37.7771	−1.21232	−0.606162	−	0.795341i	$-0.707292\pi$
$971$	−37.7771	−1.21232	−0.606162	+	0.795341i	$0.707292\pi$
$972$	0	0
$973$	−2.56231	−0.0821438
$974$	−10.3951	−0.333081
$975$	0	0
$976$	−14.5623	−0.466128
$977$	4.63932	0.148425	0.0742125	−	0.997242i	$-0.476356\pi$
$977$	4.63932	0.148425	0.0742125	+	0.997242i	$0.476356\pi$
$978$	0	0
$979$	0	0
$980$	−8.47214	−0.270632
$981$	0	0
$982$	−15.5836	−0.497292
$983$	27.3050	0.870893	0.435446	−	0.900215i	$-0.356591\pi$
$983$	27.3050	0.870893	0.435446	+	0.900215i	$0.356591\pi$
$984$	0	0
$985$	63.8328	2.03388
$986$	−4.47214	−0.142422
$987$	0	0
$988$	−16.7082	−0.531559
$989$	21.6525	0.688509
$990$	0	0
$991$	21.2705	0.675680	0.337840	−	0.941204i	$-0.390304\pi$
$991$	21.2705	0.675680	0.337840	+	0.941204i	$0.390304\pi$
$992$	−16.0344	−0.509094
$993$	0	0
$994$	10.3131	0.327111
$995$	−43.7426	−1.38674
$996$	0	0
$997$	12.9787	0.411040	0.205520	−	0.978653i	$-0.434111\pi$
$997$	12.9787	0.411040	0.205520	+	0.978653i	$0.434111\pi$
$998$	10.8541	0.343581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.s.1.1		2
3.2	odd	2		363.2.a.e.1.2		2
11.7	odd	10		99.2.f.b.82.1		4
11.8	odd	10		99.2.f.b.64.1		4
11.10	odd	2		1089.2.a.m.1.2		2
12.11	even	2		5808.2.a.bm.1.1		2
15.14	odd	2		9075.2.a.bv.1.1		2
33.2	even	10		363.2.e.h.202.1		4
33.5	odd	10		363.2.e.c.124.1		4
33.8	even	10		33.2.e.a.31.1	yes	4
33.14	odd	10		363.2.e.j.130.1		4
33.17	even	10		363.2.e.h.124.1		4
33.20	odd	10		363.2.e.c.202.1		4
33.26	odd	10		363.2.e.j.148.1		4
33.29	even	10		33.2.e.a.16.1	✓	4
33.32	even	2		363.2.a.h.1.1		2
99.7	odd	30		891.2.n.a.676.1		8
99.29	even	30		891.2.n.d.676.1		8
99.40	odd	30		891.2.n.a.379.1		8
99.41	even	30		891.2.n.d.460.1		8
99.52	odd	30		891.2.n.a.757.1		8
99.74	even	30		891.2.n.d.757.1		8
99.85	odd	30		891.2.n.a.460.1		8
99.95	even	30		891.2.n.d.379.1		8
132.95	odd	10		528.2.y.f.49.1		4
132.107	odd	10		528.2.y.f.97.1		4
132.131	odd	2		5808.2.a.bl.1.1		2
165.8	odd	20		825.2.bx.b.724.2		8
165.29	even	10		825.2.n.f.676.1		4
165.62	odd	20		825.2.bx.b.49.2		8
165.74	even	10		825.2.n.f.526.1		4
165.107	odd	20		825.2.bx.b.724.1		8
165.128	odd	20		825.2.bx.b.49.1		8
165.164	even	2		9075.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.e.a.16.1	✓	4	33.29	even	10
33.2.e.a.31.1	yes	4	33.8	even	10
99.2.f.b.64.1		4	11.8	odd	10
99.2.f.b.82.1		4	11.7	odd	10
363.2.a.e.1.2		2	3.2	odd	2
363.2.a.h.1.1		2	33.32	even	2
363.2.e.c.124.1		4	33.5	odd	10
363.2.e.c.202.1		4	33.20	odd	10
363.2.e.h.124.1		4	33.17	even	10
363.2.e.h.202.1		4	33.2	even	10
363.2.e.j.130.1		4	33.14	odd	10
363.2.e.j.148.1		4	33.26	odd	10
528.2.y.f.49.1		4	132.95	odd	10
528.2.y.f.97.1		4	132.107	odd	10
825.2.n.f.526.1		4	165.74	even	10
825.2.n.f.676.1		4	165.29	even	10
825.2.bx.b.49.1		8	165.128	odd	20
825.2.bx.b.49.2		8	165.62	odd	20
825.2.bx.b.724.1		8	165.107	odd	20
825.2.bx.b.724.2		8	165.8	odd	20
891.2.n.a.379.1		8	99.40	odd	30
891.2.n.a.460.1		8	99.85	odd	30
891.2.n.a.676.1		8	99.7	odd	30
891.2.n.a.757.1		8	99.52	odd	30
891.2.n.d.379.1		8	99.95	even	30
891.2.n.d.460.1		8	99.41	even	30
891.2.n.d.676.1		8	99.29	even	30
891.2.n.d.757.1		8	99.74	even	30
1089.2.a.m.1.2		2	11.10	odd	2
1089.2.a.s.1.1		2	1.1	even	1	trivial
5808.2.a.bl.1.1		2	132.131	odd	2
5808.2.a.bm.1.1		2	12.11	even	2
9075.2.a.x.1.2		2	165.164	even	2
9075.2.a.bv.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +2.61803 q^{5} -3.00000 q^{7} +2.23607 q^{8} -1.61803 q^{10} -1.76393 q^{13} +1.85410 q^{14} +1.85410 q^{16} +1.61803 q^{17} -5.85410 q^{19} -4.23607 q^{20} -3.47214 q^{23} +1.85410 q^{25} +1.09017 q^{26} +4.85410 q^{28} +4.47214 q^{29} +2.85410 q^{31} -5.61803 q^{32} -1.00000 q^{34} -7.85410 q^{35} +0.236068 q^{37} +3.61803 q^{38} +5.85410 q^{40} -11.9443 q^{41} -6.23607 q^{43} +2.14590 q^{46} -1.61803 q^{47} +2.00000 q^{49} -1.14590 q^{50} +2.85410 q^{52} +9.61803 q^{53} -6.70820 q^{56} -2.76393 q^{58} -10.3262 q^{59} -7.85410 q^{61} -1.76393 q^{62} -0.236068 q^{64} -4.61803 q^{65} -9.56231 q^{67} -2.61803 q^{68} +4.85410 q^{70} +5.56231 q^{71} +3.23607 q^{73} -0.145898 q^{74} +9.47214 q^{76} -9.47214 q^{79} +4.85410 q^{80} +7.38197 q^{82} +0.708204 q^{83} +4.23607 q^{85} +3.85410 q^{86} -0.527864 q^{89} +5.29180 q^{91} +5.61803 q^{92} +1.00000 q^{94} -15.3262 q^{95} -14.0344 q^{97} -1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 3 q^{5} - 6 q^{7} - q^{10} - 8 q^{13} - 3 q^{14} - 3 q^{16} + q^{17} - 5 q^{19} - 4 q^{20} + 2 q^{23} - 3 q^{25} - 9 q^{26} + 3 q^{28} - q^{31} - 9 q^{32} - 2 q^{34} - 9 q^{35}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 3 * q^5 - 6 * q^7 - q^10 - 8 * q^13 - 3 * q^14 - 3 * q^16 + q^17 - 5 * q^19 - 4 * q^20 + 2 * q^23 - 3 * q^25 - 9 * q^26 + 3 * q^28 - q^31 - 9 * q^32 - 2 * q^34 - 9 * q^35 - 4 * q^37 + 5 * q^38 + 5 * q^40 - 6 * q^41 - 8 * q^43 + 11 * q^46 - q^47 + 4 * q^49 - 9 * q^50 - q^52 + 17 * q^53 - 10 * q^58 - 5 * q^59 - 9 * q^61 - 8 * q^62 + 4 * q^64 - 7 * q^65 + q^67 - 3 * q^68 + 3 * q^70 - 9 * q^71 + 2 * q^73 - 7 * q^74 + 10 * q^76 - 10 * q^79 + 3 * q^80 + 17 * q^82 - 12 * q^83 + 4 * q^85 + q^86 - 10 * q^89 + 24 * q^91 + 9 * q^92 + 2 * q^94 - 15 * q^95 + q^97 + 2 * q^98

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